The ratio, by volume, of soap to alcohol to water in a certain solution is 2:50:100. The solution will be...

1. Translate the problem requirements: We start with \(\mathrm{soap:alcohol:water = 2:50:100}\). In the new solution, the soap-to-alcohol ratio doubles (becomes twice as large) and the soap-to-water ratio halves (becomes half as large). We need to find how much water when alcohol = 100 cc.
2. Express the new ratio relationships: Convert the doubling and halving conditions into mathematical expressions for the new ratios compared to the original ratios.
3. Set up the new solution ratios: Use the ratio relationships to determine the new \(\mathrm{soap:alcohol:water}\) ratio in the altered solution.
4. Scale to the given alcohol amount: Since the new solution contains 100 cc of alcohol, scale the entire ratio to find the corresponding amount of water.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we know and what changes are happening:

Original solution: We have a mixture where for every 2 parts soap, there are 50 parts alcohol and 100 parts water. Think of this like a recipe - if you had 2 cups of soap, you'd mix it with 50 cups of alcohol and 100 cups of water.

What's changing in the new solution:

• The ratio of soap to alcohol doubles - this means if soap was 2 and alcohol was 50 before, now soap takes up twice as much space relative to alcohol
• The ratio of soap to water halves - this means soap takes up half as much space relative to water compared to before

What we're looking for: In this new solution, we're told there will be exactly 100 cubic centimeters of alcohol, and we need to find how much water there will be.

Process Skill: TRANSLATE - Converting the ratio relationships into clear mathematical understanding

2. Express the new ratio relationships

Now let's figure out what the new ratios actually are by working through the changes step by step.

Original ratios:

• Soap to alcohol = \(\mathrm{2:50 = \frac{2}{50} = \frac{1}{25}}\)
• Soap to water = \(\mathrm{2:100 = \frac{2}{100} = \frac{1}{50}}\)

New ratios after the changes:

• New soap to alcohol ratio = \(\mathrm{2 \times (original\,soap\,to\,alcohol\,ratio) = 2 \times \frac{1}{25} = \frac{2}{25}}\)
• New soap to water ratio = \(\mathrm{\frac{1}{2} \times (original\,soap\,to\,water\,ratio) = \frac{1}{2} \times \frac{1}{50} = \frac{1}{100}}\)

Let's call the amount of soap in the new solution 'S'. Then:

• Amount of alcohol = \(\mathrm{S ÷ \frac{2}{25} = S \times \frac{25}{2} = 12.5S}\)
• Amount of water = \(\mathrm{S ÷ \frac{1}{100} = S \times 100 = 100S}\)

3. Set up the new solution ratios

From our work above, if the new solution has S parts soap, it has 12.5S parts alcohol and 100S parts water.

So our new ratio is: \(\mathrm{Soap : Alcohol : Water = S : 12.5S : 100S}\)

We can simplify this by dividing everything by S: \(\mathrm{1 : 12.5 : 100}\)

To make this easier to work with, let's multiply everything by 2 to eliminate the decimal: \(\mathrm{2 : 25 : 200}\)

Process Skill: SIMPLIFY - Converting to whole number ratios for easier calculation

4. Scale to the given alcohol amount

We now know the new solution has the ratio \(\mathrm{Soap : Alcohol : Water = 2 : 25 : 200}\)

We're told the new solution contains exactly 100 cubic centimeters of alcohol.

Since alcohol corresponds to '25' in our ratio, and we actually have 100 cc of alcohol:

Scale factor = \(\mathrm{100 ÷ 25 = 4}\)

Therefore:

• Soap = \(\mathrm{2 \times 4 = 8\,cc}\)
• Alcohol = \(\mathrm{25 \times 4 = 100\,cc}\) ✓ (matches what we're given)
• Water = \(\mathrm{200 \times 4 = 800\,cc}\)

5. Final Answer

The altered solution will contain 800 cubic centimeters of water.

Let's verify this makes sense by checking our ratio conditions:

• New soap:alcohol ratio = \(\mathrm{8:100 = 2:25 = 0.08}\)
• Original soap:alcohol ratio = \(\mathrm{2:50 = 0.04}\)
• Indeed, \(\mathrm{0.08 = 2 \times 0.04}\) ✓ (doubled as required)

• New soap:water ratio = \(\mathrm{8:800 = 1:100 = 0.01}\)
• Original soap:water ratio = \(\mathrm{2:100 = 0.02}\)
• Indeed, \(\mathrm{0.01 = \frac{1}{2} \times 0.02}\) ✓ (halved as required)

The answer is E. 800

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what "doubled" and "halved" mean in context: Students often confuse whether "the ratio of soap to alcohol is doubled" means the new ratio becomes 4:50 (doubling the soap amount) versus correctly understanding it means the proportional relationship (soap/alcohol) doubles from \(\frac{2}{50}\) to \(\frac{4}{50}\). This fundamental misunderstanding derails the entire solution.

2. Incorrectly applying ratio changes: Students may think that if the soap-to-alcohol ratio doubles and soap-to-water ratio halves, they can simply apply these changes to the original \(\mathrm{2:50:100}\) ratio independently, not realizing these changes must be consistent with a single new amount of soap.

3. Misunderstanding the constraint structure: Students may not realize that both ratio changes (doubling soap:alcohol and halving soap:water) must be satisfied simultaneously in the same solution, leading them to treat these as separate, independent conditions.

Errors while executing the approach

1. Algebraic manipulation errors with fractions: When calculating the new ratios, students frequently make errors like incorrectly computing \(\mathrm{2 \times \frac{2}{50}}\) or \(\mathrm{\frac{1}{2} \times \frac{2}{100}}\), especially when converting between ratios and fractions or when scaling to find equivalent amounts.

2. Scaling factor calculation mistakes: Once students establish the correct new ratio (like \(\mathrm{2:25:200}\)), they often make arithmetic errors when determining the scaling factor from the given alcohol amount (100 cc), such as incorrectly computing \(\mathrm{100 ÷ 25 = 4}\).

3. Unit conversion and ratio relationship errors: Students may correctly find that if soap is S, then alcohol is 12.5S and water is 100S, but then make errors when converting this to the final ratio form or when applying the scaling factor to find the actual amounts.

Errors while selecting the answer

1. Selecting intermediate calculations as the final answer: Students may correctly calculate various intermediate values (like the scaling factor of 4, or the soap amount of 8 cc) but mistakenly select one of these instead of the final water amount of 800 cc.

2. Choosing the wrong component: After correctly calculating that the new solution contains 8 cc soap, 100 cc alcohol, and 800 cc water, students may accidentally report the total volume (908 cc) or another component's volume instead of specifically the water volume requested.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose smart numbers for the initial solution

Instead of working with the abstract ratio \(\mathrm{2:50:100}\), let's use a convenient multiplier. Since we'll be doubling and halving ratios, let's multiply by 2 to get concrete amounts:

Initial solution: 4cc soap : 100cc alcohol : 200cc water

This gives us the same ratio (\(\mathrm{2:50:100}\)) but with concrete numbers that are easy to work with.

Step 2: Calculate the original individual ratios

• Original soap-to-alcohol ratio = \(\mathrm{4:100 = 1:25}\)
• Original soap-to-water ratio = \(\mathrm{4:200 = 1:50}\)

Step 3: Apply the changes to create new ratios

• New soap-to-alcohol ratio = \(\mathrm{2 \times (1:25) = 2:25 = 8:100}\)
• New soap-to-water ratio = \(\mathrm{\frac{1}{2} \times (1:50) = 1:100}\)

Step 4: Determine the new solution composition

From the new ratios:

• If alcohol = 100cc, then soap = 8cc (from 8:100 ratio)
• If soap = 8cc, then water = 800cc (from 1:100 ratio)

Step 5: Verify our answer

New solution ratio: 8cc soap : 100cc alcohol : 800cc water

This simplifies to \(\mathrm{2:25:200}\), which maintains our required ratio relationships.

Answer: 800 cubic centimeters of water